--- manual/s_phys_pkgs/text/fizhi.tex 2010/08/27 13:15:37 1.18 +++ manual/s_phys_pkgs/text/fizhi.tex 2010/08/30 23:09:21 1.19 @@ -36,11 +36,11 @@ mass flux, is a linear function of height, expressed as: \[ \pp{\eta(z)}{z} = \lambda \hspace{0.4cm}or\hspace{0.4cm} \pp{\eta(P^{\kappa})}{P^{\kappa}} = --{c_p \over {g}}\theta\lambda +-\frac{c_p}{g}\theta\lambda \] where we have used the hydrostatic equation written in the form: \[ -\pp{z}{P^{\kappa}} = -{c_p \over {g}}\theta +\pp{z}{P^{\kappa}} = -\frac{c_p}{g}\theta \] The entrainment parameter, $\lambda$, characterizes a particular subensemble based on its @@ -49,7 +49,7 @@ to the saturation moist static energy of the environment, $h^*$. Following \cite{moorsz:92}, $\lambda$ may be written as \[ -\lambda = { {h_B - h^*_D} \over { {c_p \over g} {\int_{P_D}^{P_B}\theta(h^*_D-h)dP^{\kappa}}} } , +\lambda = \frac{h_B - h^*_D}{ \frac{c_p}{g} \int_{P_D}^{P_B}\theta(h^*_D-h)dP^{\kappa}}, \] where the subscript $B$ refers to cloud base, and the subscript $D$ refers to the detrainment level. @@ -60,11 +60,11 @@ related to the buoyancy, or the difference between the moist static energy in the cloud and in the environment: \[ -A = \int_{P_D}^{P_B} { {\eta \over {1 + \gamma} } -\left[ {{h_c-h^*} \over {P^{\kappa}}} \right] dP^{\kappa}} +A = \int_{P_D}^{P_B} \frac{\eta}{1 + \gamma} +\left[ \frac{h_c-h^*}{P^{\kappa}} \right] dP^{\kappa} \] -where $\gamma$ is ${L \over {c_p}}\pp{q^*}{T}$ obtained from the Claussius Clapeyron equation, +where $\gamma$ is $\frac{L}{c_p}\pp{q^*}{T}$ obtained from the Claussius Clapeyron equation, and the subscript $c$ refers to the value inside the cloud. @@ -72,7 +72,7 @@ the clouds} is assumed to approximately balance the rate of change of $A$ {\em due to the generation by the large scale}. This is the quasi-equilibrium assumption, and results in an expression for $m_B$: \[ -m_B = {{- \left.{dA \over dt} \right|_{ls}} \over K} +m_B = \frac{- \left. \frac{dA}{dt} \right|_{ls}}{K} \] where $K$ is the cloud kernel, defined as the rate of change of the cloud work function per @@ -90,13 +90,13 @@ temperature (through latent heating and compensating subsidence) and moisture (through precipitation and detrainment): \[ -\left.{\pp{\theta}{t}}\right|_{c} = \alpha { m_B \over {c_p P^{\kappa}}} \eta \pp{s}{p} +\left.{\pp{\theta}{t}}\right|_{c} = \alpha \frac{ m_B}{c_p P^{\kappa}} \eta \pp{s}{p} \] and \[ -\left.{\pp{q}{t}}\right|_{c} = \alpha { m_B \over {L}} \eta (\pp{h}{p}-\pp{s}{p}) +\left.{\pp{q}{t}}\right|_{c} = \alpha \frac{ m_B}{L} \eta (\pp{h}{p}-\pp{s}{p}) \] -where $\theta = {T \over P^{\kappa}}$, $P = (p/p_0)$, and $\alpha$ is the relaxation parameter. +where $\theta = \frac{T}{P^{\kappa}}$, $P = (p/p_0)$, and $\alpha$ is the relaxation parameter. As an approximation to a full interaction between the different allowable subensembles, many clouds are simulated frequently, each modifying the large scale environment some fraction @@ -136,14 +136,14 @@ detrained liquid water amount given by \[ -F_{RAS} = \min\left[ {l_{RAS}\over l_c}, 1.0 \right] +F_{RAS} = \min\left[ \frac{l_{RAS}}{l_c}, 1.0 \right] \] where $l_c$ is an assigned critical value equal to $1.25$ g/kg. A memory is associated with convective clouds defined by: \[ -F_{RAS}^n = \min\left[ F_{RAS} + (1-{\Delta t_{RAS}\over\tau})F_{RAS}^{n-1}, 1.0 \right] +F_{RAS}^n = \min\left[ F_{RAS} + (1-\frac{\Delta t_{RAS}}{\tau})F_{RAS}^{n-1}, 1.0 \right] \] where $F_{RAS}$ is the instantanious cloud fraction and $F_{RAS}^{n-1}$ is the cloud fraction @@ -154,7 +154,7 @@ humidity: \[ -F_{LS} = \min\left[ { \left( {RH-RH_c \over 1-RH_c} \right) }^2, 1.0 \right] +F_{LS} = \min\left[ { \left( \frac{RH-RH_c}{1-RH_c} \right) }^2, 1.0 \right] \] where @@ -162,7 +162,7 @@ \bqa RH_c & = & 1-s(1-s)(2-\sqrt{3}+2\sqrt{3} \, s)r \nonumber \\ s & = & p/p_{surf} \nonumber \\ - r & = & \left( {1.0-RH_{min} \over \alpha} \right) \nonumber \\ + r & = & \left( \frac{1.0-RH_{min}}{\alpha} \right) \nonumber \\ RH_{min} & = & 0.75 \nonumber \\ \alpha & = & 0.573285 \nonumber . \eqa @@ -409,7 +409,7 @@ the large-scale and sub-grid scale optical depths, normalized by the total cloud fraction in the layer: -\[ \tau = \left( {F_{RAS} \,\,\, \tau_{RAS} + F_{LS} \,\,\, \tau_{LS} \over F_{RAS}+F_{LS} } \right) \Delta p, \] +\[ \tau = \left( \frac{F_{RAS} \,\,\, \tau_{RAS} + F_{LS} \,\,\, \tau_{LS} }{ F_{RAS}+F_{LS} } \right) \Delta p, \] where $F_{RAS}$ and $F_{LS}$ are the time-averaged cloud fractions associated with RAS and large-scale processes described in Section \ref{sec:fizhi:clouds}. @@ -463,11 +463,11 @@ and is written: \[ -{\dd{}{t} ({{\h} q^2})} - { \pp{}{z} ({ {5 \over 3} {{\lambda}_1} q { \pp {}{z} +{\dd{}{t} ({{\h} q^2})} - { \pp{}{z} ({ \frac{5}{3} {{\lambda}_1} q { \pp {}{z} ({\h}q^2)} })} = {- \overline{{u^{\prime}}{w^{\prime}}} { \pp{U}{z} }} - {\overline{{v^{\prime}}{w^{\prime}}} -{ \pp{V}{z} }} + {{g \over {\Theta_0}}{\overline{{w^{\prime}}{{{\theta}_v}^{\prime}}}} } -- { q^3 \over {{\Lambda} _1} } +{ \pp{V}{z} }} + {\frac{g}{\Theta_0}{\overline{{w^{\prime}}{{{\theta}_v}^{\prime}}}} +- \frac{ q^3}{{\Lambda}_1} } \] where $q$ is the turbulent velocity, ${u^{\prime}}$, ${v^{\prime}}$, ${w^{\prime}}$ and @@ -491,7 +491,7 @@ \[ K_h = \left\{ \begin{array}{l@{\quad\mbox{for}\quad}l} q \, \ell \, S_H(G_M,G_H) \, & \mbox{decaying turbulence} -\\ { q^2 \over {q_e} } \, \ell \, S_{H}(G_{M_e},G_{H_e}) \, & \mbox{growing turbulence} \end{array} \right. +\\ \frac{ q^2 }{ q_e } \, \ell \, S_{H}(G_{M_e},G_{H_e}) \, & \mbox{growing turbulence} \end{array} \right. \] and @@ -499,7 +499,7 @@ \[ K_m = \left\{ \begin{array}{l@{\quad\mbox{for}\quad}l} q \, \ell \, S_M(G_M,G_H) \, & \mbox{decaying turbulence} -\\ { q^2 \over {q_e} } \, \ell \, S_{M}(G_{M_e},G_{H_e}) \, & \mbox{growing turbulence} \end{array} \right. +\\ \frac{ q^2 }{ q_e } \, \ell \, S_{M}(G_{M_e},G_{H_e}) \, & \mbox{growing turbulence} \end{array} \right. \] where the subscript $e$ refers to the value under conditions of local equillibrium @@ -511,16 +511,18 @@ are functions of the Richardson number: \[ -{\bf RI} = { { {g \over \theta_v} \pp {\theta_v}{z} } \over { (\pp{u}{z})^2 + (\pp{v}{z})^2 } } - = { {c_p \pp{\theta_v}{z} \pp{P^ \kappa}{z} } \over { (\pp{u}{z})^2 + (\pp{v}{z})^2 } } . +{\bf RI} = \frac{ \frac{g}{\theta_v} \pp{\theta_v}{z} }{ (\pp{u}{z})^2 + (\pp{v}{z})^2 } + = \frac{c_p \pp{\theta_v}{z} \pp{P^ \kappa}{z} }{ (\pp{u}{z})^2 + (\pp{v}{z})^2 } . \] Negative values indicate unstable buoyancy and shear, small positive values ($<0.2$) indicate dominantly unstable shear, and large positive values indicate dominantly stable stratification. -Turbulent eddy diffusion coefficients of momentum, heat and moisture in the surface layer, -which corresponds to the lowest GCM level (see \ref{tab:fizhi:sigma}), +Turbulent eddy diffusion coefficients of momentum, heat and moisture in the +surface layer, which corresponds to the lowest GCM level +(see {\it --- missing table ---}%\ref{tab:fizhi:sigma} +), are calculated using stability-dependant functions based on Monin-Obukhov theory: \[ {K_m} (surface) = C_u \times u_* = C_D W_s @@ -536,12 +538,12 @@ $C_u$ is the dimensionless exchange coefficient for momentum from the surface layer similarity functions: \[ -{C_u} = {u_* \over W_s} = { k \over \psi_{m} } +{C_u} = \frac{u_* }{ W_s} = \frac{ k }{ \psi_{m} } \] where k is the Von Karman constant and $\psi_m$ is the surface layer non-dimensional wind shear given by \[ -\psi_{m} = {\int_{\zeta_{0}}^{\zeta} {\phi_{m} \over \zeta} d \zeta} . +\psi_{m} = {\int_{\zeta_{0}}^{\zeta} \frac{\phi_{m} }{ \zeta} d \zeta} . \] Here $\zeta$ is the non-dimensional stability parameter, and $\phi_m$ is the similarity function of $\zeta$ which expresses the stability dependance of @@ -551,13 +553,13 @@ $C_t$ is the dimensionless exchange coefficient for heat and moisture from the surface layer similarity functions: \[ -{C_t} = -{( {\overline{w^{\prime}\theta^{\prime}}}) \over {u_* \Delta \theta }} = --{( {\overline{w^{\prime}q^{\prime}}}) \over {u_* \Delta q }} = -{ k \over { (\psi_{h} + \psi_{g}) } } +{C_t} = -\frac{( \overline{w^{\prime}\theta^{\prime}}) }{ u_* \Delta \theta } = +-\frac{( \overline{w^{\prime}q^{\prime}}) }{ u_* \Delta q } = +\frac{ k }{ (\psi_{h} + \psi_{g}) } \] where $\psi_h$ is the surface layer non-dimensional temperature gradient given by \[ -\psi_{h} = {\int_{\zeta_{0}}^{\zeta} {\phi_{h} \over \zeta} d \zeta} . +\psi_{h} = {\int_{\zeta_{0}}^{\zeta} \frac{\phi_{h} }{ \zeta} d \zeta} . \] Here $\phi_h$ is the similarity function of $\zeta$, which expresses the stability dependance of the temperature and moisture gradients, and is specified differently for stable and unstable @@ -568,7 +570,7 @@ elements, in which temperature and moisture gradients can be quite large. Based on \cite{yagkad:74}: \[ -\psi_{g} = { 0.55 (Pr^{2/3} - 0.2) \over \nu^{1/2} } +\psi_{g} = \frac{ 0.55 (Pr^{2/3} - 0.2) }{ \nu^{1/2} } (h_{0}u_{*} - h_{0_{ref}}u_{*_{ref}})^{1/2} \] where Pr is the Prandtl number for air, $\nu$ is the molecular viscosity, $z_{0}$ is the @@ -577,7 +579,7 @@ The surface roughness length over oceans is is a function of the surface-stress velocity, \[ -{z_0} = c_1u^3_* + c_2u^2_* + c_3u_* + c_4 + {c_5 \over {u_*}} +{z_0} = c_1u^3_* + c_2u^2_* + c_3u_* + c_4 + \frac{c_5 }{ u_*} \] where the constants are chosen to interpolate between the reciprocal relation of \cite{kondo:75} for weak winds, and the piecewise linear relation of \cite{larpond:81} @@ -598,10 +600,10 @@ based functions of \cite{clarke:70}, slightly modified for the momemtum flux: \[ -{\phi_m} = { { 1 + 5 {{\zeta}_1} } \over { 1 + 0.00794 {{\zeta}_1} -(1+ 5 {{\zeta}_1}) } } \hspace{1cm} ; \hspace{1cm} -{\phi_h} = { { 1 + 5 {{\zeta}_1} } \over { 1 + 0.00794 {\zeta} -(1+ 5 {{\zeta}_1}) } } . +{\phi_m} = \frac{ 1 + 5 {{\zeta}_1} }{ 1 + 0.00794 {\zeta}_1 +(1+ 5 {\zeta}_1) } \hspace{1cm} ; \hspace{1cm} +{\phi_h} = \frac{ 1 + 5 {{\zeta}_1} }{ 1 + 0.00794 {\zeta} +(1+ 5 {{\zeta}_1}) } . \] The moisture flux also depends on a specified evapotranspiration coefficient, set to unity over oceans and dependant on the climatological ground wetness over @@ -646,7 +648,7 @@ The heat conduction through sea ice, $Q_{ice}$, is given by \[ -{Q_{ice}} = {C_{ti} \over {H_i}} (T_i-T_g) +{Q_{ice}} = \frac{C_{ti} }{ H_i} (T_i-T_g) \] where $C_{ti}$ is the thermal conductivity of ice, $H_i$ is the ice thickness, assumed to be $3 \hspace{.1cm} m$ where sea ice is present, $T_i$ is 273 degrees Kelvin, and $T_g$ is the @@ -655,11 +657,11 @@ $C_g$ is the total heat capacity of the ground, obtained by solving a heat diffusion equation for the penetration of the diurnal cycle into the ground (\cite{black:77}), and is given by: \[ -C_g = \sqrt{ {\lambda C_s \over 2\omega} } = \sqrt{(0.386 + 0.536W + 0.15W^2)2\times10^{-3} -{86400 \over 2 \pi} } \, \, . +C_g = \sqrt{ \frac{\lambda C_s }{ 2\omega} } = \sqrt{(0.386 + 0.536W + 0.15W^2)2\times10^{-3} +\frac{86400}{2\pi} } \, \, . \] -Here, the thermal conductivity, $\lambda$, is equal to $2\times10^{-3}$ ${ly\over{ sec}} -{cm \over {^oK}}$, +Here, the thermal conductivity, $\lambda$, is equal to $2\times10^{-3}$ $\frac{ly}{sec} +\frac{cm}{K}$, the angular velocity of the earth, $\omega$, is written as $86400$ $sec/day$ divided by $2 \pi$ $radians/ day$, and the expression for $C_s$, the heat capacity per unit volume at the surface, @@ -752,7 +754,7 @@ based on that derived by Pierrehumbert (1986) and is given by: \bq -|\vec{\tau}_{sfc}| = {\rho U^3\over{N \ell^*}} \left(F_r^2 \over{1+F_r^2}\right) \, \, , +|\vec{\tau}_{sfc}| = \frac{\rho U^3}{N \ell^*} \left( \frac{F_r^2}{1+F_r^2}\right) \, \, , \eq where $F_r = N h /U$ is the Froude number, $N$ is the {\em Brunt - V\"{a}is\"{a}l\"{a}} frequency, $U$ is the @@ -1374,9 +1376,9 @@ is time-averaged over its diagnostic output frequency: \[ -{\bf DIAGNOSTIC} = {1 \over TTOT} \sum_{t=1}^{t=TTOT} diag(t) +{\bf DIAGNOSTIC} = \frac{1}{TTOT} \sum_{t=1}^{t=TTOT} diag(t) \] -where $TTOT = {{\bf NQDIAG} \over \Delta t}$, {\bf NQDIAG} is the +where $TTOT = \frac{ {\bf NQDIAG} }{\Delta t}$, {\bf NQDIAG} is the output frequency of the diagnostic, and $\Delta t$ is the timestep over which the diagnostic is updated. @@ -1448,7 +1450,7 @@ through sea ice represents an additional energy source term for the ground temperature equation. \[ -{\bf QICE} = {C_{ti} \over {H_i}} (T_i-T_g) +{\bf QICE} = \frac{C_{ti}}{H_i} (T_i-T_g) \] where $C_{ti}$ is the thermal conductivity of ice, $H_i$ is the ice thickness, assumed to @@ -1490,8 +1492,8 @@ \noindent The non-dimensional stability indicator is the ratio of the buoyancy to the shear: \[ -{\bf RI} = { { {g \over \theta_v} \pp {\theta_v}{z} } \over { (\pp{u}{z})^2 + (\pp{v}{z})^2 } } - = { {c_p \pp{\theta_v}{z} \pp{P^ \kappa}{z} } \over { (\pp{u}{z})^2 + (\pp{v}{z})^2 } } +{\bf RI} = \frac{ \frac{g}{\theta_v} \pp {\theta_v}{z} }{ (\pp{u}{z})^2 + (\pp{v}{z})^2 } + = \frac{c_p \pp{\theta_v}{z} \pp{P^ \kappa}{z} }{ (\pp{u}{z})^2 + (\pp{v}{z})^2 } \] \\ where we used the hydrostatic equation: @@ -1510,15 +1512,15 @@ The surface exchange coefficient is obtained from the similarity functions for the stability dependant flux profile relationships: \[ -{\bf CT} = -{( {\overline{w^{\prime}\theta^{\prime}}}) \over {u_* \Delta \theta }} = --{( {\overline{w^{\prime}q^{\prime}}}) \over {u_* \Delta q }} = -{ k \over { (\psi_{h} + \psi_{g}) } } +{\bf CT} = -\frac{( \overline{w^{\prime}\theta^{\prime}} ) }{ u_* \Delta \theta } = +-\frac{( \overline{w^{\prime}q^{\prime}} ) }{ u_* \Delta q } = +\frac{ k }{ (\psi_{h} + \psi_{g}) } \] where $\psi_h$ is the surface layer non-dimensional temperature change and $\psi_g$ is the viscous sublayer non-dimensional temperature or moisture change: \[ -\psi_{h} = {\int_{\zeta_{0}}^{\zeta} {\phi_{h} \over \zeta} d \zeta} \hspace{1cm} and -\hspace{1cm} \psi_{g} = { 0.55 (Pr^{2/3} - 0.2) \over \nu^{1/2} } +\psi_{h} = \int_{\zeta_{0}}^{\zeta} \frac{\phi_{h} }{ \zeta} d \zeta \hspace{1cm} and +\hspace{1cm} \psi_{g} = \frac{ 0.55 (Pr^{2/3} - 0.2) }{ \nu^{1/2} } (h_{0}u_{*} - h_{0_{ref}}u_{*_{ref}})^{1/2} \] and: @@ -1540,11 +1542,11 @@ The surface exchange coefficient is obtained from the similarity functions for the stability dependant flux profile relationships: \[ -{\bf CU} = {u_* \over W_s} = { k \over \psi_{m} } +{\bf CU} = \frac{u_* }{ W_s} = \frac{ k }{ \psi_{m} } \] where $\psi_m$ is the surface layer non-dimensional wind shear: \[ -\psi_{m} = {\int_{\zeta_{0}}^{\zeta} {\phi_{m} \over \zeta} d \zeta} +\psi_{m} = {\int_{\zeta_{0}}^{\zeta} \frac{\phi_{m} }{ \zeta} d \zeta} \] \noindent $\phi_m$ is the similarity function of $\zeta$, which expresses the stability dependance of @@ -1564,9 +1566,9 @@ or moisture. In the \cite{helflab:88} adaptation of this closure, $K_h$ takes the form: \[ -{\bf ET} = K_h = -{( {\overline{w^{\prime}\theta_v^{\prime}}}) \over {\pp{\theta_v}{z}} } +{\bf ET} = K_h = -\frac{( \overline{w^{\prime}\theta_v^{\prime}}) }{ \pp{\theta_v}{z} } = \left\{ \begin{array}{l@{\quad\mbox{for}\quad}l} q \, \ell \, S_H(G_M,G_H) & \mbox{decaying turbulence} -\\ { q^2 \over {q_e} } \, \ell \, S_{H}(G_{M_e},G_{H_e}) & \mbox{growing turbulence} \end{array} \right. +\\ \frac{ q^2 }{ q_e } \, \ell \, S_{H}(G_{M_e},G_{H_e}) & \mbox{growing turbulence} \end{array} \right. \] where $q$ is the turbulent velocity, or $\sqrt{2*turbulent \hspace{.2cm} kinetic \hspace{.2cm} energy}$, $q_e$ is the turbulence velocity derived from the more simple level 2.0 model, @@ -1605,9 +1607,9 @@ In the \cite{helflab:88} adaptation of this closure, $K_m$ takes the form: \[ -{\bf EU} = K_m = -{( {\overline{u^{\prime}w^{\prime}}}) \over {\pp{U}{z}} } +{\bf EU} = K_m = -\frac{( \overline{u^{\prime}w^{\prime}} ) }{ \pp{U}{z} } = \left\{ \begin{array}{l@{\quad\mbox{for}\quad}l} q \, \ell \, S_M(G_M,G_H) & \mbox{decaying turbulence} -\\ { q^2 \over {q_e} } \, \ell \, S_{M}(G_{M_e},G_{H_e}) & \mbox{growing turbulence} \end{array} \right. +\\ \frac{ q^2 }{ q_e } \, \ell \, S_{M}(G_{M_e},G_{H_e}) & \mbox{growing turbulence} \end{array} \right. \] \noindent where $q$ is the turbulent velocity, or $\sqrt{2*turbulent \hspace{.2cm} kinetic \hspace{.2cm} @@ -1710,9 +1712,9 @@ \] where: \[ -\left.{\pp{T}{t}}\right|_{c} = R \sum_i \left( \alpha { m_B \over c_p} \Gamma_s \right)_i +\left.{\pp{T}{t}}\right|_{c} = R \sum_i \left( \alpha \frac{m_B}{c_p} \Gamma_s \right)_i \hspace{.4cm} and -\hspace{.4cm} \left.{\pp{T}{t}}\right|_{ls} = {L \over c_p } (q^*-q) +\hspace{.4cm} \left.{\pp{T}{t}}\right|_{ls} = \frac{L}{c_p} (q^*-q) \] and \[ @@ -1740,7 +1742,7 @@ \] where: \[ -\left.{\pp{q}{t}}\right|_{c} = R \sum_i \left( \alpha { m_B \over {L}}(\Gamma_h-\Gamma_s) \right)_i +\left.{\pp{q}{t}}\right|_{c} = R \sum_i \left( \alpha \frac{m_B}{L}(\Gamma_h-\Gamma_s) \right)_i \hspace{.4cm} and \hspace{.4cm} \left.{\pp{q}{t}}\right|_{ls} = (q^*-q) \] @@ -1786,11 +1788,11 @@ Finally, the net longwave heating rate is calculated as the vertical divergence of the net terrestrial radiative fluxes: \[ -\pp{\rho c_p T}{t} = - {\partial \over \partial z} F_{LW}^{NET} , +\pp{\rho c_p T}{t} = - \p{z} F_{LW}^{NET} , \] or \[ -{\bf RADLW} = \frac{g}{c_p \pi} {\partial \over \partial \sigma} F_{LW}^{NET} . +{\bf RADLW} = \frac{g}{c_p \pi} \p{\sigma} F_{LW}^{NET} . \] \noindent @@ -1821,11 +1823,11 @@ \noindent The heating rate due to Shortwave Radiation under cloudy skies is defined as: \[ -\pp{\rho c_p T}{t} = - {\partial \over \partial z} F(cloudy)_{SW}^{NET} \cdot {\rm RADSWT}, +\pp{\rho c_p T}{t} = - \p{z} F(cloudy)_{SW}^{NET} \cdot {\rm RADSWT}, \] or \[ -{\bf RADSW} = \frac{g}{c_p \pi} {\partial \over \partial \sigma} F(cloudy)_{SW}^{NET}\cdot {\rm RADSWT} . +{\bf RADSW} = \frac{g}{c_p \pi} \p{\sigma} F(cloudy)_{SW}^{NET}\cdot {\rm RADSWT} . \] \noindent @@ -1845,7 +1847,7 @@ the vertical integral or total precipitable amount is given by: \[ {\bf PREACC} = \int_{surf}^{top} \rho \Delta q_{moist} dz = - \int_{surf}^{top} \Delta q_{moist} -{dp \over g} = {1 \over g} \int_0^1 \Delta q_{moist} dp +\frac{dp}{g} = \frac{1}{g} \int_0^1 \Delta q_{moist} dp \] \\ @@ -1862,7 +1864,7 @@ the vertical integral or total precipitable amount is given by: \[ {\bf PRECON} = \int_{surf}^{top} \rho \Delta q_{cum} dz = - \int_{surf}^{top} \Delta q_{cum} -{dp \over g} = {1 \over g} \int_0^1 \Delta q_{cum} dp +\frac{dp}{g} = \frac{1}{g} \int_0^1 \Delta q_{cum} dp \] \\ @@ -1947,7 +1949,7 @@ \noindent The drag coefficient for momentum obtained by assuming a neutrally stable surface layer: \[ -{\bf CN} = { k \over { \ln({h \over {z_0}})} } +{\bf CN} = \frac{ k }{ \ln(\frac{h }{z_0}) } \] \noindent @@ -1983,7 +1985,7 @@ \noindent where \[ -\theta_{v{Nrphys+1}} = { T_g \over {P^{\kappa}_{surf}} } (1 + .609 q_{Nrphys+1}) \hspace{1cm} +\theta_{v{Nrphys+1}} = \frac{ T_g }{ P^{\kappa}_{surf} } (1 + .609 q_{Nrphys+1}) \hspace{1cm} and \hspace{1cm} q_{Nrphys+1} = q_{Nrphys} + \beta(q^*(T_g,P_s) - q_{Nrphys}) \] @@ -2014,12 +2016,12 @@ $C_g$ is obtained by solving a heat diffusion equation for the penetration of the diurnal cycle into the ground (\cite{black:77}), and is given by: \[ -C_g = \sqrt{ {\lambda C_s \over {2 \omega} } } = \sqrt{(0.386 + 0.536W + 0.15W^2)2x10^{-3} -{ 86400. \over {2 \pi} } } \, \, . +C_g = \sqrt{ \frac{\lambda C_s }{ 2 \omega } } = \sqrt{(0.386 + 0.536W + 0.15W^2)2x10^{-3} +\frac{86400.}{2\pi} } \, \, . \] \noindent -Here, the thermal conductivity, $\lambda$, is equal to $2x10^{-3}$ ${ly\over{ sec}} -{cm \over {^oK}}$, +Here, the thermal conductivity, $\lambda$, is equal to $2x10^{-3}$ $\frac{ly}{sec} +\frac{cm}{K}$, the angular velocity of the earth, $\omega$, is written as $86400$ $sec/day$ divided by $2 \pi$ $radians/ day$, and the expression for $C_s$, the heat capacity per unit volume at the surface, @@ -2161,11 +2163,11 @@ vertical divergence of the clear-sky longwave radiative flux: \[ -\pp{\rho c_p T}{t}_{clearsky} = - {\partial \over \partial z} F(clearsky)_{LW}^{NET} , +\pp{\rho c_p T}{t}_{clearsky} = - \p{z} F(clearsky)_{LW}^{NET} , \] or \[ -{\bf LWCLR} = \frac{g}{c_p \pi} {\partial \over \partial \sigma} F(clearsky)_{LW}^{NET} . +{\bf LWCLR} = \frac{g}{c_p \pi} \p{\sigma} F(clearsky)_{LW}^{NET} . \] \noindent @@ -2355,7 +2357,7 @@ the surface layer top impeded by the surface drag: \[ {\bf USTAR} = C_uW_s \hspace{1cm}where: \hspace{.2cm} -C_u = {k \over {\psi_m} } +C_u = \frac{k}{\psi_m} \] \noindent @@ -2370,7 +2372,7 @@ time from the monthly mean data of \cite{dorsell:89}. Over the ocean, the roughness length is a function of the surface-stress velocity, $u_*$. \[ -{\bf Z0} = c_1u^3_* + c_2u^2_* + c_3u_* + c_4 + {c_5 \over {u_*}} +{\bf Z0} = c_1u^3_* + c_2u^2_* + c_3u_* + c_4 + {c_5}{u_*} \] \noindent @@ -2424,11 +2426,11 @@ \noindent The heating rate due to Shortwave Radiation under clear skies is defined as: \[ -\pp{\rho c_p T}{t} = - {\partial \over \partial z} F(clear)_{SW}^{NET} \cdot {\rm RADSWT}, +\pp{\rho c_p T}{t} = - \p{z} F(clear)_{SW}^{NET} \cdot {\rm RADSWT}, \] or \[ -{\bf SWCLR} = \frac{g}{c_p } {\partial \over \partial p} F(clear)_{SW}^{NET}\cdot {\rm RADSWT} . +{\bf SWCLR} = \frac{g}{c_p } \p{p} F(clear)_{SW}^{NET}\cdot {\rm RADSWT} . \] \noindent @@ -2615,11 +2617,11 @@ \end{eqnarray*} then, since there are no surface pressure changes due to Diabatic processes, we may write \[ -\pp{T}{t}_{Diabatic} = {p^\kappa \over \pi }\pp{\pi \theta}{t}_{Diabatic} +\pp{T}{t}_{Diabatic} = \frac{p^\kappa}{\pi}\pp{\pi \theta}{t}_{Diabatic} \] where $\theta = T/p^\kappa$. Thus, {\bf DIABT} may be written as \[ -{\bf DIABT} = {p^\kappa \over \pi } \left( \pp{\pi \theta}{t}_{Diabatic} + \pp{\pi \theta}{t}_{Analysis} \right) +{\bf DIABT} = \frac{p^\kappa}{\pi} \left( \pp{\pi \theta}{t}_{Diabatic} + \pp{\pi \theta}{t}_{Analysis} \right) \] \\ @@ -2638,11 +2640,11 @@ \] then, since there are no surface pressure changes due to Diabatic processes, we may write \[ -\pp{q}{t}_{Diabatic} = {1 \over \pi }\pp{\pi q}{t}_{Diabatic} +\pp{q}{t}_{Diabatic} = \frac{1}{\pi}\pp{\pi q}{t}_{Diabatic} \] Thus, {\bf DIABQ} may be written as \[ -{\bf DIABQ} = {1 \over \pi } \left( \pp{\pi q}{t}_{Diabatic} + \pp{\pi q}{t}_{Analysis} \right) +{\bf DIABQ} = \frac{1}{\pi} \left( \pp{\pi q}{t}_{Diabatic} + \pp{\pi q}{t}_{Analysis} \right) \] \\ @@ -2656,7 +2658,7 @@ \[ {\bf VINTUQ} = \frac{ \int_{surf}^{top} u q \rho dz } { \int_{surf}^{top} \rho dz } \] -Using $\rho \delta z = -{\delta p \over g} = - {1 \over g} \delta p$, we have +Using $\rho \delta z = -\frac{\delta p}{g} = - \frac{1}{g} \delta p$, we have \[ {\bf VINTUQ} = { \int_0^1 u q dp } \] @@ -2673,7 +2675,7 @@ \[ {\bf VINTVQ} = \frac{ \int_{surf}^{top} v q \rho dz } { \int_{surf}^{top} \rho dz } \] -Using $\rho \delta z = -{\delta p \over g} = - {1 \over g} \delta p$, we have +Using $\rho \delta z = -\frac{\delta p}{g} = - \frac{1}{g} \delta p$, we have \[ {\bf VINTVQ} = { \int_0^1 v q dp } \] @@ -2706,7 +2708,7 @@ \[ {\bf VINTVT} = \frac{ \int_{surf}^{top} v T \rho dz } { \int_{surf}^{top} \rho dz } \] -Using $\rho \delta z = -{\delta p \over g} $, we have +Using $\rho \delta z = -\frac{\delta p}{g} $, we have \[ {\bf VINTVT} = { \int_0^1 v T dp } \] @@ -2771,10 +2773,10 @@ given by: \begin{eqnarray*} {\bf QINT} & = & \int_{surf}^{top} \rho q dz \\ - & = & {\pi \over g} \int_0^1 q dp + & = & \frac{\pi}{g} \int_0^1 q dp \end{eqnarray*} where we have used the hydrostatic relation -$\rho \delta z = -{\delta p \over g} $. +$\rho \delta z = -\frac{\delta p}{g} $. \\ @@ -2784,8 +2786,8 @@ \noindent The u-wind at the 2-meter depth is determined from the similarity theory: \[ -{\bf U2M} = {u_* \over k} \psi_{m_{2m}} {u_{sl} \over {W_s}} = -{ \psi_{m_{2m}} \over {\psi_{m_{sl}} }}u_{sl} +{\bf U2M} = \frac{u_*}{k} \psi_{m_{2m}} \frac{u_{sl}}{W_s} = +\frac{ \psi_{m_{2m}} }{ \psi_{m_{sl}} }u_{sl} \] \noindent @@ -2800,8 +2802,8 @@ \noindent The v-wind at the 2-meter depth is a determined from the similarity theory: \[ -{\bf V2M} = {u_* \over k} \psi_{m_{2m}} {v_{sl} \over {W_s}} = -{ \psi_{m_{2m}} \over {\psi_{m_{sl}} }}v_{sl} +{\bf V2M} = \frac{u_*}{k} \psi_{m_{2m}} \frac{v_{sl}}{W_s} = +\frac{ \psi_{m_{2m}} }{ \psi_{m_{sl}} }v_{sl} \] \noindent @@ -2816,13 +2818,13 @@ \noindent The temperature at the 2-meter depth is a determined from the similarity theory: \[ -{\bf T2M} = P^{\kappa} ({\theta* \over k} ({\psi_{h_{2m}}+\psi_g}) + \theta_{surf} ) = -P^{\kappa}(\theta_{surf} + { {\psi_{h_{2m}}+\psi_g} \over {{\psi_{h_{sl}}+\psi_g}} } -(\theta_{sl} - \theta_{surf})) +{\bf T2M} = P^{\kappa} (\frac{\theta*}{k} ({\psi_{h_{2m}}+\psi_g}) + \theta_{surf} ) = +P^{\kappa}(\theta_{surf} + \frac{ \psi_{h_{2m}}+\psi_g }{ \psi_{h_{sl}}+\psi_g } +(\theta_{sl} - \theta_{surf}) ) \] where: \[ -\theta_* = - { (\overline{w^{\prime}\theta^{\prime}}) \over {u_*} } +\theta_* = - \frac{ (\overline{w^{\prime}\theta^{\prime}}) }{ u_* } \] \noindent @@ -2838,13 +2840,13 @@ \noindent The specific humidity at the 2-meter depth is determined from the similarity theory: \[ -{\bf Q2M} = P^{\kappa} ({q_* \over k} ({\psi_{h_{2m}}+\psi_g}) + q_{surf} ) = -P^{\kappa}(q_{surf} + { {\psi_{h_{2m}}+\psi_g} \over {{\psi_{h_{sl}}+\psi_g}} } +{\bf Q2M} = P^{\kappa} \frac({q_*}{k} ({\psi_{h_{2m}}+\psi_g}) + q_{surf} ) = +P^{\kappa}(q_{surf} + \frac{ \psi_{h_{2m}}+\psi_g }{ \psi_{h_{sl}}+\psi_g } (q_{sl} - q_{surf})) \] where: \[ -q_* = - { (\overline{w^{\prime}q^{\prime}}) \over {u_*} } +q_* = - \frac{ (\overline{w^{\prime}q^{\prime}}) }{ u_* } \] \noindent @@ -2862,8 +2864,8 @@ and the model lowest level wind using the ratio of the non-dimensional wind shear at the two levels: \[ -{\bf U10M} = {u_* \over k} \psi_{m_{10m}} {u_{sl} \over {W_s}} = -{ \psi_{m_{10m}} \over {\psi_{m_{sl}} }}u_{sl} +{\bf U10M} = \frac{u_*}{k} \psi_{m_{10m}} \frac{u_{sl}}{W_s} = +\frac{ \psi_{m_{10m}} }{ \psi_{m_{sl}} }u_{sl} \] \noindent @@ -2879,8 +2881,8 @@ and the model lowest level wind using the ratio of the non-dimensional wind shear at the two levels: \[ -{\bf V10M} = {u_* \over k} \psi_{m_{10m}} {v_{sl} \over {W_s}} = -{ \psi_{m_{10m}} \over {\psi_{m_{sl}} }}v_{sl} +{\bf V10M} = \frac{u_*}{k} \psi_{m_{10m}} \frac{v_{sl}}{W_s} = +\frac{ \psi_{m_{10m}} }{ \psi_{m_{sl}} }v_{sl} \] \noindent @@ -2896,13 +2898,13 @@ temperature and the model lowest level potential temperature using the ratio of the non-dimensional temperature gradient at the two levels: \[ -{\bf T10M} = P^{\kappa} ({\theta* \over k} ({\psi_{h_{10m}}+\psi_g}) + \theta_{surf} ) = -P^{\kappa}(\theta_{surf} + { {\psi_{h_{10m}}+\psi_g} \over {{\psi_{h_{sl}}+\psi_g}} } +{\bf T10M} = P^{\kappa} (\frac{\theta*}{k} ({\psi_{h_{10m}}+\psi_g}) + \theta_{surf} ) = +P^{\kappa}(\theta_{surf} + \frac{\psi_{h_{10m}}+\psi_g}{\psi_{h_{sl}}+\psi_g} (\theta_{sl} - \theta_{surf})) \] where: \[ -\theta_* = - { (\overline{w^{\prime}\theta^{\prime}}) \over {u_*} } +\theta_* = - \frac{ (\overline{w^{\prime}\theta^{\prime}}) }{ u_* } \] \noindent @@ -2919,13 +2921,13 @@ humidity and the model lowest level specific humidity using the ratio of the non-dimensional temperature gradient at the two levels: \[ -{\bf Q10M} = P^{\kappa} ({q_* \over k} ({\psi_{h_{10m}}+\psi_g}) + q_{surf} ) = -P^{\kappa}(q_{surf} + { {\psi_{h_{10m}}+\psi_g} \over {{\psi_{h_{sl}}+\psi_g}} } +{\bf Q10M} = P^{\kappa} (\frac{q_*}{k} ({\psi_{h_{10m}}+\psi_g}) + q_{surf} ) = +P^{\kappa}(q_{surf} + \frac{\psi_{h_{10m}}+\psi_g}{\psi_{h_{sl}}+\psi_g} (q_{sl} - q_{surf})) \] where: \[ -q_* = - { (\overline{w^{\prime}q^{\prime}}) \over {u_*} } +q_* = - \frac{ (\overline{w^{\prime}q^{\prime}}) }{ u_* } \] \noindent